## Description

Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode).

States of a single oscillator correspond to \(L^2\)-normalizable functions on \(\mathbb{R}\) that have finite energy, finite variance, and finite values of all other moments (where the energy operator is defined to be the harmonic oscillator Hamiltonian); such functions form Schwartz space, a subspace of Hilbert space [1]. Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.

States can be represented by a series via a basis expansion, such as that in the countable basis of Fock states \(|n\rangle\) with \(n\geq 0\). Alternatively, states can be represented as functions over the reals by expanding in a continuous "basis" (more technically, set of tempered distributions in the space dual to Schwartz space), such as the position "basis" \(|y\rangle\) with \(y\in\mathbb{R}\) or the momentum "basis" \(|p\rangle\) with \(p\in\mathbb{R}\). States can further be represented as functions over the joint position-momentum phase space in the Wigner function formalism [2,3].

An important subset of states is formed by the Gaussian states, which are in one-to-one correspondence with a (displacement) vector and covariance matrix [4–8]. Pure Gaussian states can be obtained from the vacuum Fock state \(|n=0\rangle\) via a Gaussian unitary transformation (defined below).

## Protection

### Displacement error basis

An error set relevant to bosonic stabilizer codes is the set of displacement operators, a bosonic analogue of the Pauli string basis for qubit codes.

Displacement operators: For a single mode, its elements are products of exponentials of the mode's position and momentum operators, acting on the mode's position states \(|y\rangle\) for \(y\in\mathbb{R}\) as \begin{align} e^{-iq\hat{p}}\left|y\right\rangle =\left|y+q\right\rangle \,\,\text{ and }\,\,e^{iq\hat{x}}\left|y\right\rangle =e^{iq y}\left|y\right\rangle ~, \tag*{(1)}\end{align} where \(q\in\mathbb{R}\). The former is also called a translation, while the latter is called a modulation in signal processing. For multiple modes, error set elements are tensor products of elements of the single-oscillator error set, characterized by the vector of coefficients \(\xi\in\mathbb{R}^{2n}\).

The displacement error set is a unitary basis for bounded operators on the \(n\)-mode Hilbert space that is Dirac-orthonormal under the Hilbert-Schmidt inner product. Expanding a bounded operator in terms of displacements is called the Fourier-Weyl transform (a.k.a. Fourier-Weyl relation) [10][9; Eq. (4.11)]. For the expansion of Gaussian unitary operations in terms of displacements, see [11; Eq. (7.62)].

There are two definitions of code distance associated with displacements. The definition inherited from qubit codes is the minimum weight of a displacement operator (i.e., number of nonzero entries in \(\xi\)) that implements a nontrivial logical operation in the code. The second definition is the minimum Euclidean distance (i.e., \(\ell^2\)-norm of \(\xi\)) such that the corresponding displacement implements a nontrivial logical operation in the code.

### Loss and gain operators

An error set relevant to Fock-state bosonic codes is the set of loss operators associated with the AD channel, a common form of physical noise in bosonic systems. For a single mode, loss operators are proportional to powers of the mode's annihilation operator \(a=(\hat{x}+i\hat{p})/\sqrt{2}\), where \(\hat x\) (\(\hat p\)) is the mode's position (momentum) operator, and with the power signifying the number of particles lost during the error. For multiple modes, error set elements are tensor products of elements of the single-mode error set.

### Number-phase operators

An related error set is the set of powers of the Susskind–Glogower phase operator \(\frac{1}{\sqrt{a a^\dagger}} a\) and its adjoint [12–14] along with Fock-space rotations generated by the occupation number operator \(a^\dagger a\). These can also be obtained from qudit Pauli matrices through a limiting procedure [14] and allow one to expand trace-class operators despite not forming an orthonormal set [1]. These operators are correspong to the number-phase interpretation, a polar-like decomposition of a single mode, complementing the cartesian-like decomposition in terms of position and momentum displacements.

## Rate

## Gates

## Notes

## Parent

- Group-based quantum code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.

## Children

- Coherent-state constellation code
- Numerically optimized bosonic code
- Hybrid cat code — The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.
- Hybrid qudit-oscillator code — The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.
- Concatenated bosonic code
- Oscillator-into-oscillator code — Oscillator-into-oscillator codes are bosonic codes with an infinite-dimensional logical subspace.
- Qudit-into-oscillator code — Qudit-into-oscillator codes are bosonic codes with a finite-dimensional logical subspace.
- Bosonic stabilizer code
- Homological number-phase code — Homological number-phase codes are bosonic codes encoding logical qudits and/or logical rotors.
- Penrose tiling code — Penrose tiling codes encode information into Penrose tilings, which are non-periodic tilings of \(\mathbb{R}^n\).

## Cousins

- \(t\)-design — The notion of quantum state designs has been extended to states of a bosonic mode [53].
- Bosonic c-q code — Bosonic c-q codes are bosonic codes designed to transmit classical information.
- EA bosonic code — EA bosonic codes utilize additional ancillary modes in a pre-shared entangled state, but reduce to ordinary bosonic codes when said modes are interpreted as noiseless physical modes.
- Fermion code — Bosonic (fermionic) codes are associated with bosonic (fermionic) degrees of freedom.

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## Page edit log

- Victor V. Albert (2022-05-08) — most recent
- Victor V. Albert (2021-11-24)

## Cite as:

“Bosonic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/oscillators

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/oscillators.yml.